Departments

Cooperation and services

• Nvestigation of fundamental problems and development of theoretical bases of complex (self-organizing) systems.
• Development and research of the complex dynamics models obtained on the basis of mathematical and imitation modelling methods.
• Synthesis of new algorithms for nonlinear analysis and identification of states of complex dynamical systems.
• Creation and development of mathematical methods for modelling active media (in particular, aerohydrodynamic flows, charged particle flows in electromagnetic fields, impulse propagation on the nervous fibres, etc.) to study the general regularities origin, formation and evolution in them of self-organization structures.
• Development of the effective algorithms for modelling complex dynamics, dynamic objects in machine building, physics, chemistry, medicine, biology and ecology.
• Creation of computer technologies for modelling complex dynamical systems using modern software packages.
• Development and application of computer technologies for numerical modelling and virtual analysis of aerohydrodynamic and electrodynamic processes in the setting parts of real and designing engineering constructions.
• Esign of the software tools for computer modelling, digital processing and recognition of complex chaotic signals of various physical nature.

Results of scientific research and development

The statistical theory of gravitating cosmogonic bodies formation has been developed, and namely, within framework of this theory the models as well as evolution equations of the statistical mechanics of formation of gravitating cosmogonical bodies have been proposed:

• the known problem of gravitational condensation of infinite distributed cosmic substance is solved on the basis of the proposed statistical model of the antidiffusion processes which allows to solve the gravitational paradox for infinite homogeneous spatial substance. It is known that the Jeansʼ gravitational instability theory does not explain the reason for obtaining the critical value of the disturbation wavelength, i.e. the mechanism for increasing the wavelengths of disturbations in a stationary one-component gas medium. Within the framework of the proposed statistical model it is shown that the gravitational instability of a gas-dust cloud arises with increasing the mass density of condensation disturbations of a one-component cosmic gas medium;
• within the framework of the developed statistical model of the antidiffusion process a new law for the planetary distances distribution in our Solar system is proposed depending on the average value of the specific angular momentum of a continuous medium in the protoplanetary cloud. The proposed law of planetary distances in the Solar system generalizes the famous law of Academician O.Y. Schmidt. In the comparison with other known laws of planetary distances the new law gives a fairly good estimation of the observed planetary distances in the Solar system;
• a nonlinear time-dependent Schrödinger-like undulatory equation that describes the arising nonlinear phenomena due to the processes of self-organization in the forming cosmogonical body has been derived;
• the universal stellar law (USL) for extrasolar planetary systems connecting the temperature, size and mass of each star has been derived and justified. The analysis of the modified USL has shown that most part of the stars corresponds to the category of ideal (or classical) stars (respectively, extrasolar planetary systems) outside the concrete dependence on their belonging to the spectral classes O, B, A, F, G, K and M, although there is some error in estimating the parameters of the stars depending on their spectral belonging: a lower accuracy of the modified USL is observed for very bright stars belonging to the higher order spectral classes O, B and A, or for the more dim stars belonging to the lower spectral class M or group of red giants;
• the combined law of USL and Kepler third law for extrasolar planetary systems has been derived and justified which permits to estimate the parameters of planetary orbits.

The theory of matrix decomposition of chaotic attractors into state-space of the complex dynamical systems has been proposed. Within framework of this theory new methods of nonlinear analysis of dynamical systems have been developed:

• analytical expressions for the decomposition of operators of nonlinear dynamical systems on the basis of vector-matrix series in state-space have been derived;
• a representation of the shift operator on the trajectories of a nonlinear dynamical system through the corresponding shift operators on the trajectories of multidimensional linear dynamical systems has been proposed;
• quantitative analysis of known attractors (Lorenz, Hēnon, Rössler, Fitz-Hugh, etc.) of complex dynamic systems based on the proposed matrix series has been developed;
• a method for estimating Lyapunov characteristic exponents on the basis of matrix decomposition theory has been proposed;
• an analysis of dynamical states of the Hopfield neural network using the matrix decomposition has been carried out.

An algorithm for defining the minimal embedding dimension of a nonlinear dynamical system attractor by means of the local topological analysis of phase trajectories has been developed. The necessary and sufficient conditions for the existence of the topological stabilization for the restored attractor have been formulated. The efficiency of the proposed algorithm in the computational aspect (on average by an order) has been shown in comparison with the known correlative-topological algorithms (Grassberger-Procaccia, Schuster, etc.).
A new analytical model based on the nonlinear ordinary differential equations system describing the transition from the laminar to the turbulent regime through the periodic one has been constructed. This model also describes the dynamical fluid states arising in such transition in the Couette-Taylor hydrodynamic system (rotating coaxial cylinders). Its application for the study of aerodynamic regimes of rotor mechanisms functioning in the setting parts of machine-building constructions has been shown.

• for computer simulation of the mass (solid particles) movement process in air flows within the setting parts of mechanisms of the forage harvester KVK-800 at the “Gomselmash”, in particular “GSKB for the grain harvesting and forage harvesting equipment” (Gomel);
• for computer simulation of hydrodynamic flows with usage of rotating computational meshes arising in the working region of the hydrotransformer TGD-340A at “Amkodor”, Ltd. (Minsk);
• for computer modelling and analysis of continuous medium flows in order to improve the grain cleaning system in the grain harvester at the “GSKB for grain harvesting and forage harvesting equipment” (Gomel);
• for computer modelling of steam-water flows inside the working chamber of a microturbine and full-scale tests with the purpose of development and implementation of integrated information technology supporting the design of new types of power machines (microturbines) at “Promprivod”, Ltd. (Minsk);
• for computer modelling and analysis of particle flows in a continuous medium for the improvement of the threshing-separating device and the transportation of the grain mass in the combine harvester at the “GSKB for grain harvesting and forage harvesting equipment” (Gomel).

Developed the following software and hardware complexes:

• a set of experimental software tools for pre-processing and digital analysis of telemetric data to identify non-routine states of space on-board equipment based on the adaptive neural network classifier with the function of additional learning;
• software for the microturbine functioning parameters analysis system implemented at “Promprivod”, Ltd. (Minsk);
• a mobile robot control system as a part of working model for monitoring to reveal the facts of unauthorized intrusions.

Achievements

• Honorary Diploma of the IPI of the NAS of Belarus for the best scientific result in the field of basic research "Methods of analysis of attractors of complex dynamic systems in the phase space and their use for recognition of signals of electrocardiograms and conditions for the formation of hydrodynamic structures".
• Honorary Diploma of IPI of the National Academy of Sciences of Belarus for the best scientific result in the field of basic research “Alloriths of nonlinear analysis and recognition of speech and medical signals using the methods of entry-extractions of Volterre-Viner and the space of states”.
• Honorary Diploma of the IPI of the NAS of Belarus for the best development in the field of applied research “Modeling of air and gas flows in conjunction with the methods of synthesis and analysis of tractors in the condition space for the purpose of selling on the computer of air-gas-dynamic models in flow parts of machine-building products”.
• Honorary Diploma of the IPI of the NAS of Belarus for the best development in the field of applied research "Development of methods of interconnection of two approaches on the basis of input-enditions and the space of states to the description of complex nonlinear dynamical systems".
• Honorary Diploma of IPI of the NAS of Belarus for the best scientific result of “Methods of nonlinear analysis of chaotic attractors of complex dynamic systems based on matrix decomposition in the space of states”.
• Honorary Diploma of IIP OF THE NAS of Belarus for the most important scientific result in the field of fundamental research “Mathematical methods and computer models of self-organization of ordered spatial structures in aerohydrodynamic flows”.
• Honorary Diploma of IPI NAS of Belarus for the most important scientific result of “Methods of mathematical modeling of self-organization processes in active environments”.
• Honorary Diploma of IIP OF THE NAS of Belarus for the most important result "Satic theory of self-organization of engragging cosmogonic bodies".